LFIS–20: Definition and Fixing of the ∆ and δ Exponents

Abstract LFIS–20: ∆/δ Closure Ledger Across Nine Regimes This volume records and fixes exponent usage in Light Frame Cadence Theory (LFCT). It introduces no new parameters, observational regimes, data analyses, dynamical laws, or phase terminology. LFCT is formulated as a constraint framework rather than a field theory. It specifies representability and closure requirements on observable structure, requiring inward (TD) and outward (TS) curvature contributions to meet consistently across scales. Two related scaling exponents appear throughout the framework: ∆ (capital delta) — the cadence-balance exponent governing mass–velocity scaling in the outward (TS) contribution. δ (lowercase delta) — the induced response exponent appearing in acceleration-based observables. These exponents have appeared separately across earlier LFIS volumes, Mathematical Grounding notes, and published regime analyses, occasionally leading to ambiguity. The purpose of this volume is to consolidate definitions, fix notation explicitly, and document their relationship in a single ledger. In this volume: ∆ is derived geometrically from cadence balance, yielding the canonical value ∆ = 1/4. δ is defined as the induced response exponent δ ≡ 2∆ = 1/2. Both exponents are fixed once and are not adjusted per regime. Apparent effective exponents in empirical fits are identified as observational or population-level effects and do not redefine the canonical values. No new empirical tests are performed; all references are to previously published regime analyses. Across nine published observational regimes, the same values of ∆, δ, and the universal acceleration scale a₀ are used without tuning. Failure of LFCT would require regime-dependent exponent drift or breakdown of this closure. No such dependence is observed in the published regime suite. This volume is declarative and archival. It exists to eliminate notation drift, prevent misinterpretation of scaling exponents, and provide a stable reference for exponent commitments in LFCT as of Jan 31, 2026.This record includes both the compiled PDF and the LaTeX source used to produce it.